Commutators on Generalized Block-Triangular Algebras
Rings and Algebras
2025-10-08 v1
Abstract
The characterization of commutators in associative algebras is a classical problem in ring theory. In this paper, we address this problem for the natural class of generalized block-triangular algebras. To this end, we introduce a new invariant: the multitrace of an arbitrary element in an associative unital algebra, and prove that in a generalized block-triangular algebra, an element is a commutator if and only if its multitrace vanishes. As a consequence, we show that the set of commutators is closed under addition in these algebras. Our main result extends the classical Albert-Muckenhoupt-Shoda theorem for full matrix algebras to the broader setting of generalized block-triangular algebras.
Cite
@article{arxiv.2510.05820,
title = {Commutators on Generalized Block-Triangular Algebras},
author = {Pedro Souza Fagundes and Thiago Castilho de Mello},
journal= {arXiv preprint arXiv:2510.05820},
year = {2025}
}